基于双层位势的虚边界元计算方法
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摘要
用边界元法来求解位势问题有效而简单,但通常需要求解奇异积分,特别是当公式中有双层位势的法向导数时,会遇到超强奇异积分。若采用虚边界元法[23]就可以避开这些弱点,通过在所研究的区域之外的延拓区域中选择一个虚拟的边界,根据原问题的边界条件确定出在虚边界上分布的虚拟密度,以此来求解原区域上的解。在数值计算时,是对虚边界进行离散,由于场点和源点在不同的边界上,从而避免了传统边界元中关于奇异与超强奇异积分的处理。
通常所谓的虚边界元方法都是基于单层位势的延拓,利用在虚拟边界上的密度函数作为间接未知量,令此虚密度函数对真实边界所产生的位势及通量与边界条件一致,从而达到求解虚拟密度函数的目的[58]。
本文则利用在虚边界上分布的矩密度,得出基于双层位势的一种虚边界积分方程。作为一种间接边界积分公式,由于积分点和场点分别位于虚边界和实边界上,可以避免强奇异积分和超强奇异积分的计算,并将其应用到各种椭圆边值问题的求解上。
首先,本文针对Laplace方程的三类内边值问题建立出相应的基于双层位势的虚边界积分方程,并采用常单元来求解。其次,对广义线性Poisson类方程,本文考虑用将虚边界元和径向基函数耦合的方式来求解,对齐次方程的解采用虚边界元求解,对特解则用径向基函数逼近,从而避免区域积分项处理的繁琐。再者,进一步考虑,对双调和方程,通过引入中间变量,将其分解为两个Poisson方程,再用径向基函数同虚边界元耦合的方法求解,并应用于非线性双调和方程的求解。最后,利用自己编写的程序,通过数值算例验证了该方法的可行性和正确性。
The boundary element method is an effective numerical method for solving potential problem, but it needs to calculate singular integrals. Especially, we will meet the hyper-singular integral when we calculate the normal derivative of a double layer potential. When virtual boundary method is used,the singularity of integral can be avoided. We set a closed virtual curve, so called virtual boundary outside the domain under consideration. So the unknown virtual density function distributed on the virtual boundary can be determined by the boundary conditions given on the physical boundary via nonsingular integral equation. In calculation, since the source points are on the physical boundary, and the integral points on virtual boundary respectively, so that the integral is no longer singular or hyper-singular. The formulations used in virtual boundary are the indirect boundary integrals.
The virtual boundary method used before is based on single layer potential, in which the indirect unknown function is the density function distributed on the virtual boundary. And then the potential and its derivative on the physical boundary can be determined by virtual density function, which can be obtained according to the boundary conditions of the original problem.
In this paper, we use another kind of virtual boundary elements method, which is based on double layer potential by using of the moment density distributed on the virtual boundary. Since there is a distance between the virtual boundary and physical one, the integral is no longer singular. It can be applied into various elliptic boundary condition problems.
Firstly, we write the virtual boundary integral equation according to three boundary conditions for Laplace problem and solve it by constant elements. Secondly, for generalized Poisson equation, we combine virtual boundary method with RBFs to solve it. We use RBF approximation is employed to construct particular solution and VBCM for homogenous solution instead of domain discretization. Thirdly, for harmonic equation,we divide it into two Poisson equations by introducing intermediate function. And then, we can solve it by use of virtual boundary method and RBF, this approach can be used to solve nonlinear harmonic equation also. Finally, we write a program in Matlab and did some numerical tests, the numerical results of examples demonstrate that the scheme presented is effective and accurate.
通常所谓的虚边界元方法都是基于单层位势的延拓,利用在虚拟边界上的密度函数作为间接未知量,令此虚密度函数对真实边界所产生的位势及通量与边界条件一致,从而达到求解虚拟密度函数的目的[58]。
本文则利用在虚边界上分布的矩密度,得出基于双层位势的一种虚边界积分方程。作为一种间接边界积分公式,由于积分点和场点分别位于虚边界和实边界上,可以避免强奇异积分和超强奇异积分的计算,并将其应用到各种椭圆边值问题的求解上。
首先,本文针对Laplace方程的三类内边值问题建立出相应的基于双层位势的虚边界积分方程,并采用常单元来求解。其次,对广义线性Poisson类方程,本文考虑用将虚边界元和径向基函数耦合的方式来求解,对齐次方程的解采用虚边界元求解,对特解则用径向基函数逼近,从而避免区域积分项处理的繁琐。再者,进一步考虑,对双调和方程,通过引入中间变量,将其分解为两个Poisson方程,再用径向基函数同虚边界元耦合的方法求解,并应用于非线性双调和方程的求解。最后,利用自己编写的程序,通过数值算例验证了该方法的可行性和正确性。
The boundary element method is an effective numerical method for solving potential problem, but it needs to calculate singular integrals. Especially, we will meet the hyper-singular integral when we calculate the normal derivative of a double layer potential. When virtual boundary method is used,the singularity of integral can be avoided. We set a closed virtual curve, so called virtual boundary outside the domain under consideration. So the unknown virtual density function distributed on the virtual boundary can be determined by the boundary conditions given on the physical boundary via nonsingular integral equation. In calculation, since the source points are on the physical boundary, and the integral points on virtual boundary respectively, so that the integral is no longer singular or hyper-singular. The formulations used in virtual boundary are the indirect boundary integrals.
The virtual boundary method used before is based on single layer potential, in which the indirect unknown function is the density function distributed on the virtual boundary. And then the potential and its derivative on the physical boundary can be determined by virtual density function, which can be obtained according to the boundary conditions of the original problem.
In this paper, we use another kind of virtual boundary elements method, which is based on double layer potential by using of the moment density distributed on the virtual boundary. Since there is a distance between the virtual boundary and physical one, the integral is no longer singular. It can be applied into various elliptic boundary condition problems.
Firstly, we write the virtual boundary integral equation according to three boundary conditions for Laplace problem and solve it by constant elements. Secondly, for generalized Poisson equation, we combine virtual boundary method with RBFs to solve it. We use RBF approximation is employed to construct particular solution and VBCM for homogenous solution instead of domain discretization. Thirdly, for harmonic equation,we divide it into two Poisson equations by introducing intermediate function. And then, we can solve it by use of virtual boundary method and RBF, this approach can be used to solve nonlinear harmonic equation also. Finally, we write a program in Matlab and did some numerical tests, the numerical results of examples demonstrate that the scheme presented is effective and accurate.
引文
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[3]杨德全,赵忠生著.边界元理论及应用[M].北京:北京理工大学出版社,2002.
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[5]张鸿庆,王鸣著.有限元的数学理论[M].北京:科学出版社,1991,12.
[6]祝家麟.边界元方法—一个老而新的发展中的数值方法[J],数学的实践与认识,1983,(3):74-79.
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[8]姚寿广著,边界元数值方法及其工程应用[M].北京:国防工业出版社,1991.
[9]申光宪等著,边界元法[M].北京:机械工业出版社,1998.
[10]稽醒,臧跃龙,程玉民著,边界元法进展及通用程序[M].上海:同济大学出版社,1997.
[11]孙炳楠等著,工程中边界元法及其应用[M].浙江:浙江大学出版社,1991.
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[13]余德浩著.自然边界元方法的数学理论[M].北京:科学出版社,1993.
[14]祝家麟著.椭圆边值问题的边界元分析.[M].北京:科学出版社,1991.
[15]严重著.边界单元法基础[M].重庆:重庆大学出版社,1986.
[16]王元淳著.边界元法基础.[M].上海:上海交通大学出版社,1988.
[17]杜庆华著.边界积分方程方法-边界元法[M].北京:高等教育出版社,1989.
[18]卢盛松主编.边界元理论及应用[M].北京:高等教育出版社,1990.
[19]张凯院,高维数值积分的一种计算方法[J].工程数学学报(14)3:91-95.
[20] BREBBIA C.A. The Boundary Element Method for Engineers.[M].London: Pentech Press. 1978.
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[22]冯康著.数值计算方法[M].北京:国防工业出版社,1978.
[23]翟景春,Laplace方程边值问题的一种数值解法[J],山东科学,1995(3):21-29.
[24]陈浩,二维Laplace方程解的性质[J],工程数学,1995(12)2:95-99.
[25]祝家麟,边界元方法中的奇异性[J].重庆建筑工程学院学报1985(13)2:90—102.
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